Design and Analysis of Experiments

Transcription

1 Design and Analysis of Experiments Part VIII: Plackett-Burman, 3 k, Mixed Level, Nested, and Split-Plot Designs Prof. Dr. Anselmo E de Oliveira anselmo.quimica.ufg.br anselmo.disciplinas@gmail.com

2 Plackett-Burman Designs Two-level fractional factorial designs developed for studying k = N 1 variables in N runs, where N is a multiple of 4 Nongeometric designs (cannot be represented as cubes) Nonregular designs Regular design is one in which all effects can be estimated independently of the other effects (e.g. 2 k design) and in the case of a fractional factorial, the effects that cannot be estimated are completely aliased with the other effects (e.g. 2 k-p design) Screening Main effects are, in general, heavily confounded with two-factor interactions Plackett-Burman designs are very efficient screening designs when only main effects are of interest. Web of Science: 2017/04: : 202

3 Plus and minus signs for the Plackett-Burman Designs The design is obtained by writing down the appropriate row in the table below as a column (or row) A second column (or row) is then generated from this first one by moving the elements of the column (or row) down (or to the right) one position and placing the last element in the first position. A third column... N k Signs After the column k a row of minus sings is then added, completing the design

4 Plackett-Burman Design for N = 12, k = 11 X 1 X 2 X 3 X 4 X 5 X 6 X 7 X 8 X 9 X 10 X

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6 k = 8 N = 12 row of minus signs

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8 ? x 2

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10 > library(frf2) > design<-pb(nruns=12,nfactos=11,randomize=false) > y<-c(0.371,0.336,0.576,0.216,0.228,0.328, ,0.562,0.389,0.438,0.430,0.292) > model<-aov(y~(.), data=design) > coef(model) (Intercept) A1 B1 C D1 E1 F1 G H1 J1 K1 L

11 3 k Factorial Design Three levels: low(-1), intermediate(0), and high(+1) Regression Model y = β 0 + β 1 x 1 + β 2 x 2 + β 12 x 1 x 2 + β 11 x β 22 x 22 Possible choice to model a curvature in the response function It is not the most efficient way to model a quadratic relationship Response surface designs 2 k design augmented with central points (curvature)

12 The effects model y ijk = μ + τ i + β j + τβ ij + ε ijk The means model y ijk = μ ijk + ε ijk i = 1,2,, a j = 1,2,, b k = 1,2,, n where the mean of the ijth cell is μ ij = μ + τ i + β j + τβ ij i = 1,2,, a j = 1,2,, b k = 1,2,, n

13 In the two-factor factorial design we are interested in testing hypothesis abou the equality of row treatment effects, say H 0 : τ 1 = τ 2 = = τ a H 1 : at least one τ i 0 and the equality of column treatment effects, say H 0 : β 1 = β 2 = = β a H 1 : at least one β i 0 We are also interested in determining whether row and column treatment interact H 0 : τβ ij = 0 for all i, j H 1 : at least one τβ ij 0

14 Source of Variation Sum of Squares Degrees of Freedom Mean Square F 0 A treatments SS A a 1 SS A a 1 B treatments SS B b 1 SS B b 1 Interaction SS AB a 1 b 1 SS AB a 1 b 1 MS A MS E MS B MS E MS AB MS E Error SS E ab n 1 SS E ab n 1 Total SS T abn 1 SS A = 1 bn SS B = 1 an a i=1 b j=1 y 2 i.. y 2... abn y 2.j. y 2... abn SS AB = SS Subtotals SS A SS B SS Subtotals = 1 n a b i=1 j=1 y 2 ij. y 2... abn SS E = SS T SS A SS B SS AB SS T = a b n i=1 j=1 k=1 2 y ijk y 2... abn

15 A battery design experiment An engineer is designing a battery for use in a device that will be subjected to some extreme variations in temperature Three plate materials for the battery 3 2 factorial design Material Type Temperature ( F)

16 Material Type y.j. Temperature ( F) y i y... = 3799 SS Material = 1 bn a i=1 y 2 i.. y 2... abn = = 10, SS Temperature = = 39, SS Interaction = , , SS T = = 77, SS E = 77, , , , = 18,230.75

17 Source of Variation Sum of Squares Degrees of Freedom Mean Square F 0 P-Value Material types 10, , Temperature 39, , < Interaction 9, , Error 18, Total 77, The main effects of material type and temperature are significant; The interaction effect is also significant

18 > library(doe.base) > design<-fac.design(nlevels=3, nfactors=2, + factor.names = list(temp=c(-1,0,1), + Mat=c(-1,0,1)), + replications=4, randomize=false) > design run.no run.no.std.rp Temp Mat Blocks class=design, type= full factorial NOTE: columns run.no and run.no.std.rp are annotation, not part of the data frame

19 > y<-c(130,34,20,150,136,25,138,174,96, + 155,40,70,188,122,70,110,120,104, + 74,80,82,159,106,58,168,150,82, + 180,75,58,126,115,45,160,139,60) > design<-add.response(design=design,response=y) > summary(aov(y~temp*mat, data=design)) Df Sum Sq Mean Sq F value Pr(>F) Temp e-07 *** Mat ** Temp:Mat * Residuals Signif. codes: 0 *** ** 0.01 *

20 Tukey s Test Interaction is significant and comparisons between the means of one factor (e.g., A) may be obscured by the AB interaction One approach is to fix factor B at a specific level and apply Tukey s test to the means of factor A at that level Ex: To detect the differences among the means of the three material types: temperature level 2 (70 F) y 12. = 57.25; y 22. = ; y 32. = [XLS] Statistical Tables & Calculators T 0.05 = q 0.05,3,27 MS E n = vs 1: = vs 2: = vs 1: = = 45.47

21 > x1<-as.numeric(levels(design$temp)[design$temp]) > x2<-as.numeric(levels(design$mat)[design$mat]) > x1x1<-x1*x1 > x2x2<-x2*x2 > model<-lm(y~x1*x2+x1x1+x2x2) > summary(model) Call: lm.default(formula = y ~ x1 * x2 + x1x1 + x2x2) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) e-11 *** x e-07 *** x ** x1x x2x x1:x Signif. codes: 0 *** ** 0.01 * Residual standard error: on 30 degrees of freedom Multiple R-squared: , Adjusted R-squared: F-statistic: on 5 and 30 DF, p-value: 4.638e-06

22 Model Adequacy Checking model1<-lm(y~x1+x2, data=design) summary(model1) par(mfrow=c(2,2)) plot(model1)

23 > library(plot3d) > x<-seq(-1,1,by=.1) > y<-seq(-1,1,by=.1) > M<-mesh(x,y) > par(mar = c(2, 2, 2, 2)) > surf3d(x=m$x,y=m$y, + z= *m$x *m$y, + bty="b2",xlab="x1",ylab="x2", + zlab="y", + main=expression("y= x"[1]*" x"[2]))

24 Montgomery, pg. 398

25 > design<-fac.design(nlevels=3, nfactors=3, + factor.names = list(noozle=c(-1,0,1), + Speed=c(-1,0,1), + Pressure=c(-1,0,1)), + replications=2, randomize=false) > y<-c(-35,17,-39,-45,-65,-55,-40,20,15, + 110,55,90,-10,-55,-28,80,110,110, + 4,-23,-30,-40,-64,-61,31,-20,54, + -25,24,-35,-60,-58,-67,15,4,-30, + 75,120,113,30,-44,-26,54,44,135, + 5,-5,-55,-30,-62,-52,36,-31,4) > design<-add.response(design=design,response=y) > summary(aov(y~noozle*speed*pressure, data=design)) Df Sum Sq Mean Sq F value Pr(>F) Noozle Speed e-11 *** Pressure e-12 *** Noozle:Speed * Noozle:Pressure ** Speed:Pressure *** Noozle:Speed:Pressure Residuals Signif. codes: 0 *** ** 0.01 *

26 > x1<-as.numeric(levels(design$noozle)[design$noozle]) > x2<-as.numeric(levels(design$speed)[design$speed]) > x3<-as.numeric(levels(design$pressure)[design$pressure]) > x1x1<-x1*x1 > x2x2<-x2*x2 > x3x3<-x3*x3 > model<-lm(y~x1*x2*x3+x1x1+x2x2+x3x3) > summary(model) Call: lm.default(formula = y ~ x1 * x2 * x3 + x1x1 + x2x2 + x3x3) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) x x x x1x x2x e-10 *** x3x e-11 *** x1:x x1:x x2:x x1:x2:x Signif. codes: 0 *** ** 0.01 * Residual standard error: on 43 degrees of freedom Multiple R-squared: , Adjusted R-squared: F-statistic: on 10 and 43 DF, p-value: 8.554e-11

27 Regression Model (x 1 = nozzle, x 2 = speed, x 3 = pressure in coded units) y = x x x x 1 x x 1 x x 2 x x x x x 1 x 2 x 3 Nozzle Type (qualitative factor) 1 (x 1 = 1): y = x x x 2 x x x (x 1 = 0): y = x x x 2 x x x (x 1 = +1): y = x x x 2 x x x 3 2

28 > library(plot3d) > x<-seq(-1,1,by=.1) > y<-seq(-1,1,by=.1) > M<-mesh(x,y) > par(mar = c(2, 2, 2, 2)) > surf3d(x=m$x,y=m$y, + z=70.583*m$x*m$x *m$y*m$y, + bty="b2",xlab="x2",ylab="x3", + zlab="y", + main=expression("y=70.583x"[2]^2*" x"[3]^2))

29 > contour2d(x=x,y=y, + z=70.583*m$x*m$x *m$y*m$y, + xlab="x2",ylab="x3",lwd=3, + drape = FALSE, + main=expression("y=70.583x"[2]^2*" X"[3]^2)) The smallest observed contour is -60 (the objetive is to minimize syrup loss)

30 Factorial with Mixed Levels Two-level factorial and fractional designs are of great practical importance The three-level system is much less useful because the designs are relatively large even for a modest number of factors In some situations it is necessary to include a factor (or a few factors) that has more than two levels

31 Example Data As an example, assume that you conducted an experiment in which you were interested in the extent to which visual distraction affects younger and older people's learning and remembering. To do this, you obtained a group of younger adults and a separate group of older adults and had them learn under three conditions (eyes closed, eyes open looking at a blank field, eyes open looking at a distracting field of pictures). This is a 2 (age) x 3 (distraction condition) mixed factorial design. The scores on the data sheet below represent the number of words recalled out of ten under each distraction condition.

32 > design2x3 <- regular.design(factors=list(age=1:2,distraction=1:3), + model=~age*distraction, + nunits=24) > print(design2x3@design) Age Distraction

33 > + 7,6,6, + 8,7,6, + 7,5,4, + 6,5,2, + 5,5,4, + 5,4,3, + 6,3,2) > design2x3.aov <- aov(y~age*distraction, data=design2x3) > summary(design2x3.aov) Df Sum Sq Mean Sq F value Pr(>F) Age *** Distraction e-05 *** Age:Distraction Residuals Signif. codes: 0 *** ** 0.01 * The test indicates: a) there is a significant effect of the age condition on word memory. b) that there is a significant effect of the distraction condition on word memorization; c) the lack of an interaction between distraction and age indicates that this effect is consistent for both younger and older participants.

34 Additional Designs Nested (or Hierarchical) Designs The levels of one factor (e.g. B) are similar but not identical for different levels of another factor (e.g. A) The levels of factor B are nested under the levels of factor A Example Two-stage nested design: suppliers batches observation» A company purchases raw material from three different suppliers» Is the purity of the raw material the same from each supplier?» There are four batches of raw material available from each supplier, and three determinations of purity from each batch

35 Why it is not a factorial example?» The batches from each supplier are unique from that particular supplier Batch 1 from supplier 1 has no connection with batch 1 from supplier 2, batch 2 from supplier 1 has no connection with batch 2 from supplier 2, and so forth.» To emphasize the fact that the batches from each supplier are different batches we may rename the batches:

36 Consider a company that buys raw material in batches from three different suppliers. The purity of this raw material varies considerably, which causes problems in manufacturing the finished product. We wish to determine whether the variability in purity is attributable to differences between suppliers. Four batches of raw material are selected at random from each supplier, and three determinations of purity are made on each batch.

37 P-Values: There is no signficant effect on purity due to suppliers The purity of batches of raw material from the same supplier does differ significantly

38 Split-Plot Designs In some experiments, we may be unable to completely randomize the order of the run Whole plots or main treatments Subplots or split-plots

39 Ex: a paper manufacturer is interested in three different pulp preparation methods and four differen cooking temperatures to study these effects on the tensile strength of the paper Each replicate requires 12 observations, and the experimenter has decided to run 3 replicates 36 runs A batch of pulp is produced by one of the three methods This batch is divided into four samples, and each sample is cooked at one of the four temperatures Then a second batch of pulp is made up using another of the three methods...

40 Why it is not a factorial example (three levels of preparation method, factor A, four levels of temperature, factor B)? If this is the case, then the order of the experimentation within each replicate should be completely randomized Treatment combinations (preparation method and temperature) should be randomly selected But, the experimenter did no collect the data this way He made up a batch of pulp and obtained observations for all temperatures from that batch The only feasible way to run this experiment: preparing the bathes (economics) and the size of the batches A completely randomized factorial experiment: 36 batches of pulp Split-plot design: 9 batches total

41 9 whole plots (preparation methods) 4 subplot treatments (temperatures)

42 The linear model y ijk = μ + τ i + β j + τβ ij + γ k + τγ ik + βγ jk + τβγ ijk + ε ijk i = 1,2,, r j = 1,2,, a k = 1,2,, b where τ i, β j, and τβ ij represent the whole plot and correspond, respectively, to replicates, main treatments (factor A), and whole-plot error (replicates A); and γ k, τγ ik, βγ jk, and τβγ ijk represent the subplot and correspond, respectively, to the subplot treatment (factor B), the replicates B and AB interactions, and the subplot error (replicates AB).

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45 The split-plot design has an agricultural heritage Whole plots: large areas of land Subplots: smaller areas of land within large areas Ex: several varieties of a crop could be planted in different fields (whole plots), one variety to a field Each field could be divided into, say, four subplots, and each subplot could be treated with different type os fertilizer. Here the crop varietes are the main treatments and the different fertilizers are the subtreatments.

46 Lab